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Solution of inverse anomalous diffusion problems using empirical and phenomenological models

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Abstract

In recent years, the anomalous diffusion phenomenon has attracted attention from the scientific community due to the number of applications and development of experimental procedures to observe this phenomenon. From a mathematical point of view, anomalous diffusion can be described by empirical (algebraic models) and phenomenological models (differential models with integer or fractional order). Considering the empirical models, one of the ways to observe the presence of anomalous diffusion is to check if the particle concentration profiles collapse around the reference profile, defined in terms of a new independent variable that depends on spatial and temporal variables. This new variable (scale factor) is used to characterize the diffusion type (classical or anomalous). For phenomenological models, diffusivity is usually considered as a function of particle concentration to characterize anomalous diffusion. For both cases, an inverse problem needs to be formulated and solved to obtain the parameters for each methodology. In this context, the present contribution aims to formulate and solve two inverse anomalous diffusion problems related to empirical and phenomenological models. For this purpose, two experimental data sets and Differential Evolution (DE) are considered. The results obtained demonstrate that the DE strategy was able to find good estimates for the scale factor and for the parameters related to the phenomenological model.

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Acknowledgements

The author would like to acknowledge the financial support from FAPEMIG and CNPq. Dr. William would like to thank the Programa de Pós-Graduação em Modelagem e Otimização daUniversidade Federal de Goiás, Regional Catalão for the support during his post doctorate in the Programa Nacional de Pós-Doutorado (PNPD)-CAPES.

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Correspondence to Fran Sérgio Lobato.

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Lima, W.J., Lobato, F.S. & de Oliveira Arouca, F. Solution of inverse anomalous diffusion problems using empirical and phenomenological models. Heat Mass Transfer 55, 3053–3063 (2019). https://doi.org/10.1007/s00231-019-02642-w

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